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Limits and Derivatives · Question 5

JEE Main 2026 (Online) 2nd April Morning ShiftMCQMedium📈 Topic in 10% papers+4−1
1

If x 2 (x3-5x2+ax+b ) ( x-1-1 ) e(x-1)=m, then a+b+m is equal to :

  1. 5
  2. 6
  3. 8
  4. 10

Answer

(2) 6

Solution

Put x=2+h, h 0:

x-1-1= 1+h-1= h 1+h+1 h2 and e(x-1)= e(1+h) h

∴ Denominator h22= (x-2)22

Let p(x)=x3-5x2+ax+b. For a finite limit, p(x) 0 like (x-2)2, so (x-2)2 divides p(x):

p(2)=0 8-20+2a+b=0 2a+b=12

p'(2)=0 12-20+a=0 a=8, b=-4

p(x)=x3-5x2+8x-4=(x-1)(x-2)2

∵ p(x) p(x) as p(x) 0:

m= x 2 (x-1)(x-2)2(x-2)2/2=2(2-1)=2

∴ a+b+m=8-4+2=6

Answer: (2)

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